In the system of equations above, the latent vector \(\mathbf {x}_{t,M}\)  is present in both (REF ) and (). Also, \(\mathbf {x}_{t,P}\)  is present in both () and (). In order to make sure that the values of \(\mathbf {x}_{t,M}\)  and \(\mathbf {x}_{t,P}\)  are equivalent for the different equations when fitting the joint model, \(\mathbf {x}_{t,M}\)  in (REF ) is assumed to be an (almost) identical copy of \(\mathbf {x}_{t,M}\)  in (). Similarly, \(\mathbf {x}_{t,P}\)  in () is assumed to be an (almost) identical `copy' of \(\mathbf {x}_{t,P}\)  in () where \(\alpha _1\)  is a scaling parameter. To create these `copies', the latent field \(\mathbf {x}_t\)  is extended to \(\mathbf {\chi }_t = \begin{pmatrix} \mathbf {x}_t^{\intercal } & \mathbf {x}^{*\intercal }_t \end{pmatrix}^\intercal , t=1,\ldots ,T\) , where \(\mathbf {x}^*_t = \begin{pmatrix} \mathbf {x}^{*\intercal }_{t,M} & \mathbf {x}^{*\intercal }_{t,P} \end{pmatrix}^\intercal \)  is a copy of \(\mathbf {x}_t\) . The prior specification for the extended latent field at time \(t\) , \(\pi (\mathbf {\chi }_t)\) , will ensure that \(\mathbf {x}^*_t\)  is an identical copy of \(\mathbf {x}_t\) . In particular, \(\mathbf {x}^{*}_{t,M}\)  and \(\mathbf {x}^{*}_{t,P}\)  will be defined later in such a way that \(\operatorname{\mathbb {E}}(\mathbf {x}^{*}_{t,M}) = \mathbf {x}_{t,M}\)  and \(\operatorname{\mathbb {E}}(\mathbf {x}^{*}_{t,P}) = \alpha _1 \mathbf {x}_{t,P}\) , where \(\alpha _1\)  is a scaling parameter. This is the same approach as in [1]} and [2]} and is called a data augmentation approach.
